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Chapter 4 / Exercise 17
Intermediate Algebra Within Reach
Larson
Expert Verified
1 Table of Contents Preliminaries 2.1 Velocities and Tangents 2.2 The Limit of a Function 2.3 Calculating Limits Using Limit Laws 2.4 The Precise Definition of Limit. 2.5 Continuity 2.6 Limits at infinity. 2.7 The Derivatives and Rates of Change 2.8, 3.1 The Derivative as a Function and Differentiation Rules. 3.2 The Product and Quotient Rules 3.3 Derivatives of Trigonometric Functions 3.4 The Chain Rule 3.5 Implicit Differentiation 3.6 Derivatives of Logarithmic Functions 3.7 Derivatives in the Natural and Social Sciences 3.9 Related Rates 4.1 Maximum and Minimum Values 4.3 What Derivatives Tell Us About The Shape Of A Graph. 4.5 Summary of Curve Sketching 4.7 Optimization 4.8 Newton’s Method 4.2 Rolle’s Theorem and The Mean Va lue Theorem
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Chapter 4 / Exercise 17
Intermediate Algebra Within Reach
Larson
Expert Verified
2 Preliminaries FATAL FLAWS: If you make any of the errors that appear on the following list, you will automatically lose all credit on the problem in which it is made. Flaws may be added to this list as the quarter progresses. A fatal flaw occurs when you indicate otherwise. 1. (? ± ?) ? ≠ ? ? ± ? ? The most frequent occurrence of this error is to claim that (? + ?) 2 = ? 2 + ? 2 when if fact (? + ?) 2 = ? 2 + 2?? + ? 2 , which is obtained by foiling the factor of (a + b) with itself or by using a result memorized from some previous class. 2. √? ? + ? ? 𝑛 ≠ √? ? 𝑛 + √? ? 𝑛 or a + b. Most commonly, this error is seen as √? 2 + ? 2 = ? + ?. 3. ?+?? ? ≠ ? + ? We can correctly cancel any value that is a factor of every term in the numerator and every term of the denominator. For example: ? 2 + ?? ?? = ? ? ? + ? ? = 1 ∗ ? + ? ? 4. ? ?+? ? ? + ? ? Notice that ? ? + ? ? = ?? ?? + ?? ?? = ??+?? ?? ? ?+? We can write: ?+? ? = ? ? + ? ? .
3 Rational and Negative Exponents In general: √? ? 𝑛 = ? ? ? and ? −? = 1 ? 𝑛 . Also, (? ? ) ? = ? ?? = (? ? ) ? Examples: √? = √? 2 = ? 1 2 √? 3 5 = ? 3 5 For example, 8 3 5 = √8 3 5 = (√8 5 ) 3 = 2 5 = 32 ? 2 3 = 1 √? 2 3 , i. e. , 8 2 3 = 1 4 Exercises: Convert the following radical expressions to positive rational exponents. ? 3 = ______ √? 2 3 =______ ? 2 3 6 = ______ 1 ? −1/3 = ______ 1 √? −4 3 =______ Factoring and Multiplication In general, ? 2 − ? 2 = (? + ?)(? − ?) (? ± ?) 2 = ? 2 ± 2?? + ? 2 , (? + ?) 3 = ? 3 + 3? 2 ? + 3?? 2 + ? 3 ? 3 ± ? 3 = (? ± ?)(? 2 ∓ ?? + ? 2 ) Examples: ? 2 − 1 = (? + 1)(? − 1) ? 2 − 4 = (? + 2)(? − 2) ? 2 − 5 = (? + √5 )(? − √5 ) ? − 4 = (√? + 2)(√? − 2) (? + 5) 2 = ? 2 + (2)(?)(5) + 5 2 = ? 2 + 10? + 25 (−2? + 3) 2 = (−2?) 2 + (2)(−2?)(3) + 3 2 = 4? 2 − 12? + 9
4 Everyone can factor an expression like 3? 5 − 6? 4 + 12? 2 = 3? 2 (? 3 − 2? 2 + 4) . For the coefficients, we find the greatest common factor and for the base, in this case ? , we factor out the smallest power of ? , which is ? 2 in this example. With expressions involving negative or rational exponents, the process is exactly the same. Examples: ? 3 2 − ? 1 2 = ? 1 2 (? − 1) Note that the smallest exponent is 1 2 ⁄ . ? −3 2 − ? 1 2 = ? −3 2 (1 − ? 2 ) Note that the smallest exponent is −3 2 ⁄ . 16? −4 5 + 4? 1 5 = 4? −4 5 (4 + ?) Note that the smallest exponent is −4 5 ⁄ . Exercises: Factor the following and check your answers by multiplying the factors. 4? 2 − 25 = ______________________ 6? 2 − 7? + 1 = ____________________________ ? 2 − 8? + 16 = _________________________________ ? 2 + 9 = _________________________________ ? 2 3 − ? 1 3 = _____________________________ 5? 3 4 − 10? −1 4 = _____________________________ Lines The slope of the line through two points is m = Δ? Δ? = ? 1 − ? 0 ? 1 − ? 0 . Given the slope m and a point (? 0 , ? 0 ) , the equation of the line through the point is ? − ? 0 = m(? − ? 0 ).